摘要：数字图像处理与分析领域里的一个重要研究方向是对图像形状特征的表达。我们可以从一幅数字图像计算而得到的一系列矩值，这些矩值通常表达了图像形状的全局特征，由矩函数可以得到一系列的不变量，它在图像发生平移，缩放，或旋转变换后，仍然保持不变，从而可以作为目标的特征向量被广泛地应用于图像的识别中。此外，如何用数学的手段构造更好，更实用的矩来满足目前图像分析的需要仍是有待解决的问题。非正交连续矩、径向连续正交矩、直角连续正交矩以及直接离散正交矩是最具有代表性的四种矩。本文通过讨论此四种矩来研究和比较各种矩的缩放不变量的构造。

关键词:矩; 不变量; 缩放

Abstract: In digital image processing and analysis domain，an important research direction is the expression method of  an image shape. We can obtain a series of moment from a digital image, which usually express the overall features of an image. Then a series of invariants can achieved from these moments. They can keep invariant under the transform of translation, scale and rotation. Hence, invariants of moments were widely applied as the global features in the image recognition. In addition, how uses mathematics method to construct a better, a more practical moment is still a pending solution question. The non-orthogonal continual moment, the radial direction continual orthogonal moment, the orthogonal continual moment and the discrete orthogonal moment is the most representative four kinds of moments. This article discusses these four kinds of moments, and we compare three kinds of scale invariants of them. 

Key words: Geometric Moment; Zernike Moments; Legendre Moments; Tchebichef Moments, Invariant of moments; scale invariants;